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Unformatted text preview: 6.207/14.15: Networks Lecture 12: Applications of Game Theory to Networks Daron Acemoglu and Asu Ozdaglar MIT October 21, 2009 1 Networks: Lecture 12 Introduction Outline Trac equilibrium: the Pigou example General formulation with single origin-destination pair Multi-origin-destination trac equilibria Congestion games and atomic trac equilibria Potential functions and potential games Network cost-sharing Strategic network formation. Reading: EK, Chapter 8. Jackson, Chapter 6. 2 Networks: Lecture 12 Introduction Motivation Many games are played over networks, in the sense that players interact with others linked to them through a network-like structure. Alternatively, in several important games, the actions of players correspond to a path in a given network. The most important examples are choosing a route in a trac problem or in a data routing problem. Other examples are cost sharing in network-like structures. Finally, the formation of networks is typically a game-theoretic (strategic) problem. In this lecture, we take a first look at some of these problems, focusing on trac equilibria and formation of networks. 3 Networks: Lecture 12 Wardrop Equilibria The Pigou Example of Trac Equilibrium Recall the following simple example from lecture 9, where a unit mass of trac is to be routed over a network: no congestion effects delay depends on congestion 1 unit of traffic System optimum (minimizing aggregate delay) is to split trac equally between the two routes, giving 1 1 3 min C system ( x S ) = l i ( x i S ) x i S = + = . x 1+ x 2 1 4 2 4 i Instead, the Nash equilibrium of this large (non-atomic) game, also referred to as Wardrop equilibrium , is x 1 = 1 and x 2 = (since for any x 1 &lt; 1, l 1 ( x 1 ) &lt; 1 = l 2 (1 x 1 )), giving an aggregate delay of C eq ( x WE ) = l i ( x i WE ) x i WE = 1 + 0 = 1 &gt; 3 . 4 i 4 Networks: Lecture 12 Wardrop Equilibria The Wardrop Equilibrium Why the Wardrop equilibrium? It is nothing but a Nash equilibrium in this game, in view of the fact that it is non-atomiceach player is infinitesimal. Thus, taking the strategies of others as given is equivalent to taking aggregates, here total trac on different routes, as given. Therefore, the Wardrop equilibrium (or the Nash equilibrium of a large game) is a convenient modeling tool when each participant in the game is small. A small technical detail: so far we often took the set of players, I , to be a finite set. But in fact nothing depends on this, and in non-atomic games, I is typically taken to be some interval in R , e.g., [0 , 1]. 5 Networks: Lecture 12 Wardrop Equilibria More General Trac Model Let us now generalize the Pigou example. In the general model, there are several origin-destination pairs and multiple paths linking these pairs 2 2 2 2 3x 1 0 0 0 0 x+1 Cost = 2x1 + 2x3=8 6 Networks: Lecture 12 Wardrop Equilibria More General Trac Model: Notation Let us start with a single origin-destination pair....
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